Problem Statement
Find the derivative of $f(x) = \sin x$ from first principles.
Verified Solution & Marking Scheme
State definition of derivative from first principles
$$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} = \lim_{h \to 0} \frac{\sin(x + h) - \sin x}{h}$$
Use sum-to-product formula sin C - sin D
$$\sin(x + h) - \sin x = 2 \cos\left( \frac{2x + h}{2} \right) \sin\left( \frac{h}{2} \right)$$
$$f'(x) = \lim_{h \to 0} \frac{2 \cos\left( x + \frac{h}{2} \right) \sin\left( \frac{h}{2} \right)}{h} = \lim_{h \to 0} \cos\left( x + \frac{h}{2} \right) \cdot \lim_{h \to 0} \frac{\sin(h/2)}{h/2}$$
Evaluate standard limits
Since $$\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1$$, we have:
$$f'(x) = \cos(x + 0) \cdot (1) = \cos x$$